Figure 2.2 Kepler s second law. The areas A1 and A2 swept out in unit time are equal.
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Orbits and Launching Methods
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consequence of this is that the satellite takes longer to travel a given distance when it is farther away from earth. Use is made of this property to increase the length of time a satellite can be seen from particular geographic regions of the earth. 2.4 Kepler s Third Law Kepler s third law states that the square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. The mean distance is equal to the semimajor axis a. For the artificial satellites orbiting the earth, Kepler s third law can be written in the form a3 n2 (2.2)
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where n is the mean motion of the satellite in radians per second and is the earth s geocentric gravitational constant. Its value is (see Wertz, 1984, Table L3). 3.986005 10
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Equation (2.2) applies only to the ideal situation of a satellite orbiting a perfectly spherical earth of uniform mass, with no perturbing forces acting, such as atmospheric drag. Later, in Sec. 2.8, the effects of the earth s oblateness and atmospheric drag will be taken into account. With n in radians per second, the orbital period in seconds is given by P 2 n (2.4)
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The importance of Kepler s third law is that it shows there is a fixed relationship between period and semimajor axis. One very important orbit in particular, known as the geostationary orbit, is determined by the rotational period of the earth and is described in Chap. 3. In anticipation of this, the approximate radius of the geostationary orbit is determined in the following example.
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Calculate the radius of a circular orbit for which the period is 1 day.
There are 86,400 seconds in 1 day, and therefore the mean motion is n 2 86400 7.272 10
From Kepler s third law: a c 3.986005 (7.272 1014 10 5)2 d
42,241 km Since the orbit is circular the semimajor axis is also the radius.
2.5 De nitions of Terms for Earth-Orbiting Satellites As mentioned previously, Kepler s laws apply in general to satellite motion around a primary body. For the particular case of earth-orbiting satellites, certain terms are used to describe the position of the orbit with respect to the earth. Subsatellite path. This is the path traced out on the earth s surface directly below the satellite. Apogee. The point farthest from earth. Apogee height is shown as ha in Fig. 2.3. Perigee. The point of closest approach to earth. The perigee height is shown as hp in Fig. 2.3.
Figure 2.3 Apogee height ha, perigee height hp, and inclination i. la is the line of apsides.
Orbits and Launching Methods
Line of apsides. The line joining the perigee and apogee through the center of the earth. Ascending node. The point where the orbit crosses the equatorial plane going from south to north. Descending node. The point where the orbit crosses the equatorial plane going from north to south. Line of nodes. The line joining the ascending and descending nodes through the center of the earth. Inclination. The angle between the orbital plane and the earth s equatorial plane. It is measured at the ascending node from the equator to the orbit, going from east to north. The inclination is shown as i in Fig. 2.3. It will be seen that the greatest latitude, north or south, reached by the subsatellite path is equal to the inclination. Prograde orbit. An orbit in which the satellite moves in the same direction as the earth s rotation, as shown in Fig. 2.4. The prograde orbit is also known as a direct orbit. The inclination of a prograde orbit always lies between 0 and 90 . Most satellites are launched in a prograde orbit because the earth s rotational velocity provides part of the orbital velocity with a consequent saving in launch energy.
Prograde and retrograde orbits.
Retrograde orbit. An orbit in which the satellite moves in a direction counter to the earth s rotation, as shown in Fig. 2.4. The inclination of a retrograde orbit always lies between 90 and 180 . Argument of perigee. The angle from ascending node to perigee, measured in the orbital plane at the earth s center, in the direction of satellite motion. The argument of perigee is shown as w in Fig. 2.5. Right ascension of the ascending node. To define completely the position of the orbit in space, the position of the ascending node is specified. However, because the earth spins, while the orbital plane remains stationary (slow drifts that do occur are discussed later), the longitude of the ascending node is not fixed, and it cannot be used as an absolute reference. For the practical determination of an orbit, the longitude and time of crossing of the ascending node are frequently used. However, for an absolute measurement, a fixed reference in space is required. The reference chosen is the first point of Aries, otherwise known as the vernal, or spring, equinox. The vernal equinox occurs when the sun crosses the equator going from south to north, and an imaginary line drawn from this equatorial crossing through the center of the sun points to the first point of Aries (symbol ). This is the line of Aries. The right ascension of the ascending node is then the angle measured eastward, in the equatorial plane, from the line to the ascending node, shown as in Fig. 2.5. Mean anomaly. Mean anomaly M gives an average value of the angular position of the satellite with reference to the perigee. For a